Problem: Find the coefficient of the $x^2$ term in the expansion of the product $(ax^3 + 3x^2 - 2x)(bx^2 - 7x - 4)$.
Solution: We only need to worry about the terms that multiply to have a degree of $2$. This would be given by the product of the terms $3x^2$ and $-4$ as well as the product of the terms $-2x$ and $-7x$. Thus, $$(3x^2) \times (-4) + (-2x) \times (-7x) = -12x^2 + 14x^2 = 2x^2,$$and the coefficient is $\boxed{2}$.